CLASSROOM ACTIVITY TO DEVELOP RELATIONAL THINKING

During the development of student understanding of the equal sign as relational equivalence, the students engaged in two major representations. Some students considered balance equations as two equations separated by the equal sign (Carpenter et al., 2005b). For example, Mike reflectively wrote: My thinking was challenged…instead of thinking 7+5 ≠ 3+9 [seven plus five does not equal three plus nine]… and thinking 3 was the answer but seeing that there were two problems 7 + 5 and 3 + 9. Students with this view use computation to solve open number sentence problems (Carpenter et al., 2003; M. Stephens, 2006). For example, Peter justified 8 + 6 = 9 + 5: True…because eight plus six equals fourteen and nine plus five equals fourteen. In a more sophisticated model, students are able to solve the problems by using the relation between both expressions without carrying out a calculation (Jacobs et al., 2007; M. Stephens). This relational type of thinking is recognised as a precursor to formal algebraic thinking (M. Stephens). Consequently, a focus of the study was to develop student use of this form of thinking.

True and false number sentences and open number sentence equations proved useful to develop relational strategies. Initially, instruction tasks which focused on number sentences which used closely related numbers proved effective. During small group work Hannah voiced: Eleven minus four is the same as ten minus three…because you’re just taking away

one more away from the eleven than the ten. Rani later in a whole group discussion used the difference of one as a justification: If you do eleven minus four…both of these two numbers they are just one number higher than these ones. If you have a number like twelve minus five and then there was thirteen take away six, they would both be the same and you don’t have to subtract the numbers to find out if it’s true.

Specific teacher actions were also important to support the students towards developing confidence with relational strategies. During the whole group discussion, Ella selected a student to model the use of a relational strategy to prove the number sentence was true:

Steve: Seven plus five equals three plus nine… equals is like a set of scales… on one side there’s seven plus five and the other it’s three plus nine and they have to weigh the same thing. If you changed the actual numbers that were there, [draws scales] if you minused two off the five and then you plused the two you took off the five on to the seven you’d get nine and three so it would be the same because if you’ve nine and three on one side and nine and three on the other side it would be the same.

Ella consistently pressed students beyond calculating the answer to justify using relational strategies. Before commencing an activity she indicated: I want you to think whether there is a way to prove without actually adding up the numbers. When noting the use of computation she prompted the students: I don't want you to have to add the numbers up so think about how can you look at the statement and think is it true or false. When a computational strategy was used in a whole group discussion she probed for a relational strategy: is there a way that you can show seven plus five is the same as three plus nine without actually adding it up? Just have some thinking time about how you can prove that without actually adding up those numbers. Ella highlighted the efficiency of the relational approaches: So you’re telling us that you didn’t have to subtract the numbers on both

sides? You just looked at the number? Right I want everyone looking at Rani because this is really important. To make explanations more accessible, Ella modelled how relational strategies could be recorded using arrows (see Figure 5.1).

Figure 5.1 Recording relational strategies using arrows

She encouraged students to be explicit in their explanations. For example, Rani explained her solution strategy for 256 + 3 = 246 + 13:

Rani: From the two hundred and forty-six to the two hundred and fifty-six there is ten there and from the three to the thirteen there is ten there as well.

Ella: Are you adding or subtracting that ten?... Talk to the person next to you about whether or not it is adding or subtracting the ten?...

Rani: Subtracting ten and that's adding ten.

Relational strategies, however, require time to develop and the students fluctuated between their use of calculational and relational strategies. In an early lesson, Matthew gave a relational explanation for 498 + 12 = 488 + 22: Four hundred and ninety-eight plus twelve is the same as four hundred and eighty-eight plus twenty-two because twelve and twenty- two are pretty much the same except that has ten more [points to 22] and that has ten less than that [points to 12] …take the ten off that [points to 498] to make it four hundred and eighty-eight and then adding that ten onto twelve so it will be the same. However, in a later lesson he preferred to use computation to solve the number sentence 256 + 3 = 246 + 13: I think it is true because if you added a three onto two hundred and fifty-six that would equal two hundred and fifty-nine and if you added a thirteen onto two hundred and forty-six that would equal two hundred and fifty-nine as well. The students also drew on computational strategies when they encountered difficulties convincing their peers through relational explanation. In one example, Rachel began using a relational strategy to solve 583 – 529 = 83 - 29: You’d take away the five hundreds…if you look at it carefully…you go five hundred

and eighty-three minus five hundred and twenty-nine is the same as eighty-three minus twenty-nine…take away the five hundreds on those and then it will be eighty-three take away twenty-nine is the same as eighty-three take away twenty-nine. When Rani disagreed her group began to calculate the answers:

Rani: If you take away five hundred and eighty-three from five hundred and twenty- nine it will be a higher number than that…maybe if we took away this first.

Rachel: So you take that away.

Rani: Eighty-three round that and the closest number to that is eighty.

Matthew: …The closest number to that is thirty…

Rani: Eighty minus thirty is fifty.

5.2.4 CLASSROOM ACTIVITY TO ENRICH UNDERSTANDING OF